Review of Quantitative Finance and Accounting

, Volume 45, Issue 3, pp 463–483 | Cite as

R-2GAM stochastic volatility model: flexibility and calibration

  • Cheng-Few Lee
  • Oleg SokolinskiyEmail author
Original Research


This paper investigates the potential of the 2GAM stochastic volatility model for capturing varying properties of option prices represented by the implied volatility surface. The 2GAM model is shown to be a generalization of the Heston model. Then, taking the original Heston model as the benchmark, the paper explores the flexibility allowed by the 2GAM model. More precisely, the focus is on the restricted 2GAM (R-2GAM) model which builds upon the Heston model reproducing a given short-term implied volatility skew. Going from theory to practice, the paper suggests a numerically-feasible calibration procedure for the R-2GAM model. In an application to the valuation of the S&P 500 option contracts this paper addresses the challenges of calibrating the R-2GAM model to market prices and raises concerns of possible over-parameterization.


Stochastic volatility Implied volatility smile Calibration 

JEL Classification



  1. Andersen LBG, Piterbarg VV (2010) Interest rate modeling, volume I: foundation and vanilla models. Atlantic Financial Press, LondonGoogle Scholar
  2. Ang JS, Jou KD, Lai TY (2013) A comparison of formulas to compute implied standard deviation. In: Lee CF, Lee AC (eds) Encyclopedia of finance, Springer, Berlin, pp 765–776CrossRefGoogle Scholar
  3. Bakshi G, Cao C, Chen Z (1997) Empirical performance of alternative option pricing models. J Financ 52:2003–2049. doi: 10.1111/j.1540-6261.1997.tb02749.x CrossRefGoogle Scholar
  4. Beckers S (1980) The constant elasticity of variance model and its implications for option pricing. J Financ 35:661–673. doi: 10.1111/j.1540-6261.1980.tb03490.x CrossRefGoogle Scholar
  5. Binder JJ, Merges MJ (2001) Stock market volatility and economic factors. Rev Quant Financ Account 17:5–26. doi: 10.1023/A:1011207919894 CrossRefGoogle Scholar
  6. Black F, Scholes M (1973) The valuation of options and corporate liabilities. J Polit Econ 81:637–654CrossRefGoogle Scholar
  7. Chan KC, Karolyi GA, Longstaff FA, Sanders AB (1992) An empirical comparison of alternative models of the short-term interest rate. J Financ 47:1209–1227. doi: 10.1111/j.1540-6261.1992.tb04011.x CrossRefGoogle Scholar
  8. Chang J-R, Hung M-W, Lee C-F, Lu H-M (2007) The jump behavior of foreign exchange market: analysis of Thai Baht. Rev Pac Basin Financ Mark Pol 10:265–288. doi: 10.1142/S0219091507001069 CrossRefGoogle Scholar
  9. Chen R-R, Lee C-F, Lee H-H (2009) Empirical performance of the constant elasticity variance option pricing model. Rev Pac Basin Financ Mark Pol 12:177–217. doi: 10.1142/S0219091509001605 CrossRefGoogle Scholar
  10. Christoffersen P, Heston S, Jacobs K (2006) Option valuation with conditional skewness. J Econom 131:253–284. doi: 10.1016/j.jeconom.2005.01.010 CrossRefGoogle Scholar
  11. Christoffersen P, Heston S, Jacobs K (2009) The shape and term structure of the index option smirk: why multifactor stochastic volatility models work so well. Manag Sci 55:1914–1932CrossRefGoogle Scholar
  12. Cox J (1975) Notes on option pricing I: constant elasticity of diffusions. Unpublished note. Graduate School of Business. Stanford UniversityGoogle Scholar
  13. Derman E, Kani I (1994) Riding on a smile. Risk 7:32–39Google Scholar
  14. Dumas B, Fleming J, Whaley RE (1998) Implied volatility functions: empirical tests. J Financ 53:2059–2106. doi: 10.1111/0022-1082.00083 CrossRefGoogle Scholar
  15. Dupire B (1994) Pricing with a smile. Risk 7:18–20Google Scholar
  16. Eisenberg L, Jarrow R (1994) Option pricing with random volatilities in complete markets. Rev Quant Financ Account 4:5–17. doi: 10.1007/BF01082661 CrossRefGoogle Scholar
  17. Engle R (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50:987–1007CrossRefGoogle Scholar
  18. Eraker B (2004) Do stock prices and volatility jump? Reconciling evidence from spot and option prices. J Financ 59:1367–1403. doi: 10.1111/j.1540-6261.2004.00666.x CrossRefGoogle Scholar
  19. Gatheral J (2006) The volatility surface: a practitioner’s guide. Wiley, Hoboken, NJGoogle Scholar
  20. Glasserman P (2003) Monte Carlo methods in financial engineering. Springer, BerlinCrossRefGoogle Scholar
  21. Harikumar T, de Boyrie ME, Pak SJ (2004) Evaluation of Black–Scholes and GARCH models using currency call options data. Rev Quant Financ Account 23:299–312. doi: 10.1023/B:REQU.0000049318.78363.3c CrossRefGoogle Scholar
  22. Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6:327–343. doi: 10.1093/rfs/6.2.327 CrossRefGoogle Scholar
  23. Heston SL, Nandi S (2000) A closed-form GARCH option valuation model. Rev Financ Stud 13:585–625. doi: 10.1093/rfs/13.3.585 CrossRefGoogle Scholar
  24. Hull JC, White A (1987) The pricing of options on assets with stochastic volatilities. J Financ 42:281–300. doi: 10.1111/j.1540-6261.1987.tb02568.x CrossRefGoogle Scholar
  25. Jones CS (2003) The dynamics of stochastic volatility: evidence from underlying and option markets. J Econom 116:181–224. doi: 10.1016/S0304-4076(03)00107-6 CrossRefGoogle Scholar
  26. Lo CC, Skindilias K (2013) Local volatility calibration during turbulent periods. Rev Quant Financ Account 1–20. doi: 10.1007/s11156-013-0412-6
  27. Manaster S, Koehler G (1982) The calculation of implied variances from the Black–Scholes model: a note. J Financ 37:227–230. doi: 10.1111/j.1540-6261.1982.tb01105.x CrossRefGoogle Scholar
  28. Medvedev AN, Scaillet O (2004) A simple calibration procedure of stochastic volatility models with jumps by short term asymptotics. Research paper HEC, Genève and FAME, Université de GenèveGoogle Scholar
  29. Mozumder S, Sorwar G, Dowd K (2013) Option pricing under non-normality: a comparative analysis. Rev Quant Financ Account 40:273–292. doi: 10.1007/s11156-011-0271-y CrossRefGoogle Scholar
  30. Rahman S, Lee C-F, Ang KP (2002) Intraday return volatility process: evidence from NASDAQ stocks. Rev Quant Financ Account 19:155–180. doi: 10.1023/A:1020683012149 CrossRefGoogle Scholar
  31. Rebonato R (2004) Volatility and correlation: the perfect hedger and the fox. 2nd edn. Wiley, LondonCrossRefGoogle Scholar
  32. Ritchken P, Trevor R (1999) Pricing options under generalized GARCH and stochastic volatility processes. J Financ 54:377–402. doi: 10.1111/0022-1082.00109 CrossRefGoogle Scholar
  33. Scott LO (1987) Option pricing when the variance changes randomly: theory, estimation, and an application. J Financ Quant Anal 22:419–438. doi: 10.2307/2330793 CrossRefGoogle Scholar
  34. Schroder M (1989) Computing the constant elasticity of variance option pricing formula. J Financ 44:211–219. doi: 10.1111/j.1540-6261.1989.tb02414.x CrossRefGoogle Scholar
  35. Shreve S (2004) Stochastic calculus for finance II: continuous-time models. 1st edn. Springer, BerlinGoogle Scholar
  36. Wiggins JB (1987) Option values under stochastic volatilities. J Financ Econ 19:351–372. doi: 10.1016/0304-405X(87)90009-2 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Rutgers Business School - Newark and New BrunswickPiscatawayUSA

Personalised recommendations